On queueing in coded networks queue size follows degrees of freedom

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On queueing in coded networks queue size
follows degrees of freedom

• Muriel Médard
• joint work with
• Jay Kumar Sundararajan and Devavrat Shah
  

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Problem addressed

• Time is slotted
• Stochastic arrivals into an infinite-capacity
  buffer
• Goal
• Send every packet to every receiver, while
  keeping the queue occupancy small

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Problem addressed
Known With network coding, we can get to
capacity - In every slot, send a linear
combination that is innovative to all receivers
(except those that have already caught up with
sender)?
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Problem addressed
Focus of this work Effectively manage the queue
at the sender, so that its size tracks the
backlog in degrees of freedom, and not the number
of undelivered packets
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Related earlier work

• Stability of the backlog in degrees of freedom
  (virtual queues)?
• Ho-Viswanathan '05 Backpressure algorithms for
  intra-session coding
• Eryilmaz-Lun '07 Multiple unicasts with
  inter-session coding
• Effect of finite buffer Lun et al. '06
• Queueing delay of a multicast erasure channel
• Shrader-Ephremides '06 Delay analysis in a
  block-based coding scheme

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Our Contribution

• New queueing mechanism proposed streaming, not
  block-based
• Queue size tracks backlog in linear degrees of
  freedom (virtual queue size)?
• Implications
• Traditional queueing theoretic analysis of
  virtual queues extends to the actual physical
  memory used
• Stability of virtual queue carries over to actual
  queue

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Main Idea
Vector spaces representing knowledge V Knowledge
of sender VjKnowledge of receiver j V? Common
knowledge of all receivers

• Store linear combinations of original packets
• No need to store information commonly known at
  all receivers (i.e. V?)

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Algorithm Outline

• In each time slot,
• Using the feedback, compute Vjs and hence V?.
• Update V to include new arrivals
• Compute a basis B? for V?.
• Extend this basis to a basis B for V.
• Replace current contents of queue with linear
  combinations of packets whose coefficient vectors
  are those in B\B?.

V Knowledge of sender VjKnowledge of receiver
j V? Common knowledge of all receivers
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Queueing Uncoded vs. Coded
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Main Relation

• QPhysical queue size
• LHS the amount of the sender's knowledge that
  is not known at all receivers
• RHS Sum of backlogs in linear degrees of
  freedom of all the receivers

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Conclusions and Future Work

• Proposed scheme connects the physical memory used
  at the sender to the backlog in linear degrees of
  freedom this allows queueing results on virtual
  queues to be extended to physical queues
• It reduces the amount of storage used at the
  sender. This is beneficial if multiple streams
  share buffer memory
• Future
• Use feedback to minimize decoding delay by
  careful choice of linear combinations
• Bound the overhead by bounding the coding window
  size